Hudson was eligible for salary arbitration and given that he has missed most of the past two seasons, the D-backs were unwilling to tender him a contract and risk him getting a raise over the $518,500 he made last season.
Hudson made just nine starts in 2012 before having Tommy John surgery and last June, just weeks before he was set to return, he reinjured his elbow during a Minor League rehab start and had to have a second Tommy John surgery.
When the D-backs non-tendered him, Hudson became a free agent, but both sides have said they would still like to get a deal done.
"I think we kind of ran out of time," D-backs GM Kevin Towers said referring to last Monday night's deadline. "I think that we were more than likely -- I can't speak for the other side -- but I would like to say that we were at the five-yard line or inside of that. So, my hope is that we will come to a resolution and he will be a Diamondback in the very near future."
Hudson has continued to rehab with D-backs head athletic trainer Ken Crenshaw and the training staff at the team's Salt River Fields Spring Training facility this week.
"I feel so comfortable with Ken and everybody I've been working with for the past two years," Hudson said. "I don't really want to go anywhere else. Obviously if something doesn't work out then we'll have to figure something else out, but as long as we're cordial and we're talking and everything, I really want to make sure I stay working out with those guys."
The team faced a similar situation recently with reliever Matt Reynolds, who injured his elbow last June and had Tommy John surgery in September. Reynolds is expected to miss most, if not all, of the 2014 season and he, too, was eligible for salary arbitration.
Reynolds and the D-backs agreed in October on a one-year $550,000 contract for 2014 with a $600,000 club option for 2015.
Hudson declined to get into the specifics of what the two sides had discussed.
"We had been talking back and forth for a couple of days, just talking about what each side wanted," Hudson said of the few days before the deadline. "Honestly, they had so much going on with hiring the coaching staff that the deadline kind of snuck up on both of us, I think. Then all of a sudden it was like, 'Oh no, we've got to figure this out.' We just couldn't quite get it done. I have no hard feelings or animosity or anything. I realize I'm a guy coming off two Tommy John surgeries and the market is not great for guys like me with the injury history the last couple of years, but at the same time I wanted to be sure I was still getting somewhat compensated for what I had done in the past. That was my biggest thing."
Hudson has been a starter since coming to the big leagues, but given two Tommy John surgeries, a switch to the bullpen might be the most likely path his future takes.
While he said he expected to get something done with Hudson in a timely fashion, Towers did not want to put a timetable on when Hudson might be able to return to action.